In mathematics, in the area of wavelet analysis, a refinable function is a function which fulfils some kind of self-similarity. A function φ {\displaystyle \varphi } is called refinable with respect to the mask h {\displaystyle h} if
φ ( x ) = 2 ⋅ ∑ k = 0 N − 1 h k ⋅ φ ( 2 ⋅ x − k ) {\displaystyle \varphi (x)=2\cdot \sum _{k=0}^{N-1}h_{k}\cdot \varphi (2\cdot x-k)}
This condition is called refinement equation, dilation equation or two-scale equation. Using the convolution (denoted by a star, *) of a function with a discrete mask and the dilation operator D {\displaystyle D} one can write more concisely:
φ = 2 ⋅ D 1 / 2 ( h ∗ φ ) {\displaystyle \varphi =2\cdot D_{1/2}(h*\varphi )}
It means that one obtains the function, again, if you convolve the function with a discrete mask and then scale it back. There is a similarity to iterated function systems and de Rham curves. The operator φ ↦ 2 ⋅ D 1 / 2 ( h ∗ φ ) {\displaystyle \varphi \mapsto 2\cdot D_{1/2}(h*\varphi )} is linear. A refinable function is an eigenfunction of that operator. Its absolute value is not uniquely defined. That is, if φ {\displaystyle \varphi } is a refinable function, then for every c {\displaystyle c} the function c ⋅ φ {\displaystyle c\cdot \varphi } is refinable, too. These functions play a fundamental role in wavelet theory as scaling functions.
Properties
Values at integral points A refinable function is defined only implicitly. It may also be that there are several functions which are refinable with respect to the same mask. If φ {\displaystyle \varphi } shall have finite support and the function values at integer arguments are wanted, then the two scale equation becomes a system of simultaneous linear equations. Let a {\displaystyle a} be the minimum index and b {\displaystyle b} be the maximum index of non-zero elements of h {\displaystyle h} , then one obtains
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